Definition:Zu Chongzhi Fraction
Definition
The Zu Chongzhi fraction is an exceptionally accurate approximation to $\pi$ (pi):
- $\pi \approx \dfrac {355} {113}$
whose decimal expansion is:
- $\pi \approx 3 \cdotp 14159 \, 292$
This sequence is A068079 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Also known as
The Zu Chongzhi fraction is rendered variously according to the transliteration of his name into Latin characters: Tsu Ch'ung-Chi's fraction is an example.
The fraction $\dfrac {355} {113}$ is also called Metius' number, for Adriaan Metius, who discovered it independently in around the $16$th century.
Source of Name
This entry was named for Zu Chongzhi.
Historical Note
The Zu Chongzhi fraction $\dfrac {355} {113}$ as an approximation for $\pi$ (pi) was derived by Zu Chongzhi and his son Zu Geng.
Adriaan Metius fortuitously rediscovered it independently around the $16$th century.
He did this by taking mediant of two limits $\dfrac {377} {120}$ and $\dfrac {333} {106}$ calculated by his father. This is guaranteed to generate a number between those limits, but the usefulness of the approximation was lucky.
Sources
- 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$